Chromatic Homotopy Theory

This blog-post is dedicated to this day, $\pi$-day (14th of march), where we celebrate $\pi_*$, the stable homotopy groups. As has been the case a couple of times already, when faced with an increased workload I tend to neglect writing on this blog. It is only natural that increased amounts of work in one section lead to a decreased amount of work in another — there is, after all, only a finite amount of time given to us. But, for the remainder of my PhD I will solely focus on research and outreach, hence I will hopefully have some more time to write and think. This post has been a long time coming and features the precise area of mathematics where I do most of my research, namely exotic algebraic models. I will throughout this post, and its sequels, explain what these are and connect it to almost all the previous blog posts I have made for the last couple of years. This will also set up some of the needed background for presenting my own research, which I will do once I am done writing the paper presenting it. ...

In a recent lunch conversation with Nils Baas we, among a plethora of other things, discussed the chromatic redshift phenomenon in stable homotopy theory. Nils was explaining some things about the results he had published together with Bjørn Dundas and John Rognes on using $2$-vector bundles to explain the algebraic K-theory of topological K-theory, i.e. $K(ku)$. This spectrum has height $2$, and since $ku$ has height $1$ this exhibits a phenomenon called redshift. The word redshift is used due to the word “chromatic” used for the height filtration of the stable homotopy category. Since we increase the height by one, we in some sense get our electro-magnetic frequency (which in this analogy is chromatic height) “shifted” towards the red end of the spectrum. This phenomena has been studied for many years and is the background for one of the more important long-standing conjectures in chromatic homotopy theory, namely the chromatic redshift conjecture. This conjecture roughly states that the behaviour exhibited above by $ku$ is not specific to $ku$. More specifically: the algebraic K-theory of a spectrum shifts the height by $1.$ Nils knew of but had not read the recent paper proving the last piece of the puzzle of this conjecture. The paper in question is titled “The chromatic nullstellensatz” and is a beastly paper of over a hundred pages containing mostly highly technical proofs and computations. I sent him the arXiv link on email and added my short thoughts about the proof. After sending it I realized I could expand a bit upon the comments I made to him about the proof and post it as a blog post. I have after all claimed that I want to be better at writing and publishing blog-posts, as well as produce some shorter posts. While digging into the proof and trying to expand upon the comments I made the post turned a bit longer and more technical than planned. Perhaps this is good, as it reflects the immense technicality and complexity of both the research area, conjecture and the proof. Let me just state before we start that there are details of the proofs and explanations I have swept under the rug — a rug that contains much of the actual difficulties and technicalities. I do not claim to understand everything in the proof; any wrong interpretation or wrong idea is on me and not the authors in any of the papers mentioned. ...

Hi, long time no see. I used to be very persistent about writing at least one blog-post each month, but as one can see, I have taken a break for a couple months due to vacation and increased teaching and lecturing duties at NTNU. Incidentally it coincided exactly with the 2 year mark of having posted at least once a month, often more. I have been writing stuff, but not anything worth posting. Anyway, I wanted to get into the flow of posting once a month again, as it really helps me with learning and focusing on certain topics Im interested in, and I have been contacted by several people who find the posts illuminating and helpful for understanding a topic. This post is a long time coming, as I started writing it in March… Anyway, lets get to some mathematics, but before that I want to mention that the cover image is generated by Dall-E 2 using the prompt “periodic derived category painting”. ...